Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) は、Springer-Verlagが出版している大学院レベルの数学教科書シリーズです。このシリーズは、数学の様々な分野を網羅し、高度な数学的知識を学ぶための標準的な教材として、世界中の研究者や学生に広く利用されています。

シリーズの特徴

GTMシリーズは、その一貫したフォーマットと質の高さで知られています。書籍は、標準的なサイズの黄色い表紙で、上部が白くデザインされているため、容易に識別できます。内容は、学部レベルの教科書シリーズであるUndergraduate Texts in Mathematics (UTM) よりも高度な内容を扱っている傾向にありますが、両シリーズ間には内容や難易度において重複する部分も存在します。そのため、GTMは必ずしも大学院生のみを対象としているわけではなく、学部生が高度な数学を学ぶため、あるいは大学院進学を目指す学生にとっても有益な教材となる場合があります。

一部のGTM書籍は日本語に翻訳され、丸善出版から出版されています。これにより、日本語を母語とする学習者もGTMシリーズの恩恵を受けることができます。ただし、GTMシリーズは多岐にわたる分野をカバーしており、全てが翻訳されているわけではありません。

内容

GTMシリーズは、代数学、幾何学、解析学、トポロジー、数理物理学など、数学の幅広い分野をカバーしています。各書籍は、特定の分野の基礎から応用までを体系的に解説しており、大学院レベルの研究に必要な厳密な数学的議論を展開しています。GTMシリーズは、単なる教科書としてだけでなく、研究の参考書としても活用されており、研究者にとっても非常に価値の高いシリーズです。

代表的な書籍例

以下に、GTMシリーズの代表的な書籍をいくつか紹介します。

• - Introduction to Axiomatic Set Theory
• - Measure and Category
• - Topological Vector Spaces
• - A Course in Homological Algebra
• - Categories for the Working Mathematician
• - Projective Planes
• - A Course in Arithmetic
• - Axiomatic Set Theory
• - Introduction to Lie Algebras and Representation Theory
• - A Course in Simple-Homotopy Theory
• - Functions of One Complex Variable I
• - Advanced Mathematical Analysis
• - Rings and Categories of Modules
• - Stable Mappings and Their Singularities
• - Lectures in Functional Analysis and Operator Theory
• - The Structure of Fields
• - Random Processes
• - Measure Theory
• - A Hilbert Space Problem Book
• - Fibre Bundles
• - Linear Algebraic Groups
• - An Algebraic Introduction to Mathematical Logic
• - Linear Algebra
• - Geometric Functional Analysis and Its Applications
• - Real and Abstract Analysis
• - Algebraic Theories
• - General Topology
• - Commutative Algebra I
• - Commutative Algebra II
• - Lectures in Abstract Algebra I: Basic Concepts
• - Lectures in Abstract Algebra II: Linear Algebra
• - Lectures in Abstract Algebra III: Theory of Fields and Galois Theory
• - Differential Topology
• - Principles of Random Walk
• - Several Complex Variables and Banach Algebras
• - Linear Topological Spaces
• - Mathematical Logic
• - Several Complex Variables
• - An Invitation to C*-Algebras
• - Denumerable Markov Chains
• - Modular Functions and Dirichlet Series in Number Theory
• - Linear Representations of Finite Groups
• - Rings of Continuous Functions
• - Elementary Algebraic Geometry
• - Probability Theory I
• - Probability Theory II
• - Geometric Topology in Dimensions 2 and 3
• - General Relativity for Mathematicians
• - Linear Geometry
• - Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory
• - A Course in Differential Geometry
• - Algebraic Geometry
• - A Course in Mathematical Logic for Mathematicians
• - Combinatorics with Emphasis on the Theory of Graphs
• - Introduction to Operator Theory I: Elements of Functional Analysis
• - Algebraic Topology: An Introduction
• - Introduction to Knot Theory
• - p-adic Numbers, p-adic Analysis, and Zeta-Functions
• - Cyclotomic Fields
• - Mathematical Methods of Classical Mechanics
• - Elements of Homotopy Theory
• - Fundamentals of the Theory of Groups
• - Graph Theory - An Introductory Course
• - Fourier Series - A Modern Introduction Volume 1
• - Differential Analysis on Complex Manifolds
• - Introduction to Affine Group Schemes
• - Local Fields
• - Linear Operators in Hilbert Spaces
• - Cyclotomic Fields II
• - Singular Homology Theory
• - Riemann Surfaces
• - Classical Topology and Combinatorial Group Theory
• - Algebra
• - Multiplicative Number Theory
• - Basic Theory of Algebraic Groups and Lie Algebras
• - Algebraic Geometry - An Introduction to Birational Geometry of Algebraic Varieties
• - Lectures on the Theory of Algebraic Numbers
• - A Course in Universal Algebra
• - An Introduction to Ergodic Theory
• - A Course in the Theory of Groups
• - Lectures on Riemann Surfaces
• - Differential Forms in Algebraic Topology
• - Introduction to Cyclotomic Fields
• - A Classical Introduction to Modern Number Theory
• - Fourier Series - A Modern Introduction Volume 2
• - Introduction to Coding Theory
• - Cohomology of Groups
• - Associative Algebras
• - Introduction to Algebraic and Abelian Functions
• - An Introduction to Convex Polytopes
• - The Geometry of Discrete Groups
• - Sequences and Series in Banach Spaces
• - Modern Geometry — Methods and Applications Part I: The Geometry of Surfaces, Transformation Groups, and Fields
• - Foundations of Differentiable Manifolds and Lie Groups
• - Probability-1
• - A Course in Functional Analysis
• - Introduction to Elliptic Curves and Modular Forms
• - Representations of Compact Lie Groups
• - Finite Reflection Groups
• - Harmonic Analysis on Semigroups - Theory of Positive Definite and Related Functions
• - Galois Theory
• - Lie Groups, Lie Algebras, and Their Representations
• - Complex Analysis
• - Modern Geometry — Methods and Applications Part II: The Geometry and Topology of Manifolds
• - SL2(R)
• - The Arithmetic of Elliptic Curves
• - Applications of Lie Groups to Differential Equations
• - Holomorphic Functions and Integral Representations in Several Complex Variables
• - Univalent Functions and Teichmüller Spaces
• - Algebraic Number Theory
• - Elliptic Curves
• - Elliptic Functions
• - Brownian Motion and Stochastic Calculus
• - A Course in Number Theory and Cryptography
• - Differential Geometry: Manifolds, Curves and Surfaces
• - Measure and Integral — Volume 1
• - Algebraic Groups and Class Fields
• - Analysis Now
• - An Introduction to Algebraic Topology
• - Weakly Differentiable Functions — Sobolev Spaces and Functions of Bounded Variation
• - Cyclotomic Fields I and II
• - Theory of Complex Functions
• - Numbers
• - Modern Geometry — Methods and Applications Part III: Introduction to Homology Theory
• - Complex Variables — An Introduction
• - Linear Algebraic Groups
• - A Basic Course in Algebraic Topology
• - Partial Differential Equations
• - Representation Theory
• - Tensor Geometry — The Geometric Viewpoint and its Uses
• - A First Course in Noncommutative Rings
• - Iteration of Rational Functions — Complex Analytic Dynamical Systems
• - Algebraic Geometry
• - Coding and Information Theory
• - Advanced Linear Algebra
• - Algebra — An Approach via Module Theory
• - Harmonic Function Theory
• - A Course in Computational Algebraic Number Theory
• - Topology and Geometry
• - Optima and Equilibria
• - Gröbner Bases — A Computational Approach to Commutative Algebra
• - Real and Functional Analysis
• - Measure Theory
• - Noncommutative Algebra
• - Homology Theory — An Introduction to Algebraic Topology
• - Computability — A Mathematical Sketchbook
• - Algebraic K-Theory and Its Applications
• - An Introduction to the Theory of Groups
• - Foundations of Hyperbolic Manifolds
• - Commutative Algebra — with a View Toward Algebraic Geometry
• - Advanced Topics in the Arithmetic of Elliptic Curves
• - Lectures on Polytopes
• - Algebraic Topology — A First Course
• - An Introduction to Analysis
• - Quantum Groups
• - Classical Descriptive Set Theory
• - Integration and Probability
• - Field Theory
• - Functions of One Complex Variable II
• - Differential and Riemannian Manifolds
• - Polynomials and Polynomial Inequalities
• - Groups and Representations
• - Permutation Groups
• - Additive Number Theory The Classical Bases
• - Additive Number Theory: Inverse Problems and the Geometry of Sumsets
• - Differential Geometry — Cartan's Generalization of Klein's Erlangen Program
• - Field and Galois Theory
• - Combinatorial Convexity and Algebraic Geometry
• - Matrix Analysis
• - Sheaf Theory
• - Riemannian Geometry
• - Classical Topics in Complex Function Theory
• - Graph Theory
• - Foundations of Real and Abstract Analysis
• - An Introduction to Knot Theory
• - Introduction to Riemannian Manifolds
• - Analytic Number Theory
• - Nonsmooth Analysis and Control Theory
• - Banach Algebra Techniques in Operator Theory
• - A Course on Borel Sets
• - Numerical Analysis
• - Ordinary Differential Equations
• - An Introduction to Banach Space Theory
• - Modern Graph Theory
• - Using Algebraic Geometry
• - Fourier Analysis on Number Fields
• - Moduli of Curves
• - Lectures on the Hyperreals
• - Lectures on Modules and Rings
• - Problems in Algebraic Number Theory
• - Fundamentals of Differential Geometry
• - Elements of Functional Analysis
• - Advanced Topics in Computational Number Theory
• - One-Parameter Semigroups for Linear Evolution Equations
• - Elementary Methods in Number Theory
• - Basic Homological Algebra
• - The Geometry of Schemes
• - A Course in p-adic Analysis
• - Theory of Bergman Spaces
• - An Introduction to Riemann-Finsler Geometry
• - Diophantine Geometry
• - Introduction to Topological Manifolds
• - The Symmetric Group — Representations, Combinatorial Algorithms, and Symmetric Functions
• - Galois Theory
• - Rational Homotopy Theory
• - Problems in Analytic Number Theory
• - Algebraic Graph Theory
• - Analysis for Applied Mathematics
• - A Short Course on Spectral Theory
• - Number Theory in Function Fields
• - Algebra
• - Lectures on Discrete Geometry
• - From Holomorphic Functions to Complex Manifolds
• - Partial Differential Equations
• - Algebraic Functions and Projective Curves
• - Matrices — Theory and Applications
• - Model Theory: An Introduction
• - Introduction to Smooth Manifolds
• - The Arithmetic of Hyperbolic 3-Manifolds
• - Smooth Manifolds and Observables
• - Convex Polytopes
• - Lie Groups, Lie Algebras, and Representations - An Elementary Introduction
• - Fourier Analysis and its Applications
• - Metric Structures in Differential Geometry
• - Lie Groups
• - Spaces of Holomorphic Functions in the Unit Ball
• - Combinatorial Commutative Algebra
• - A First Course in Modular Forms
• - The Geometry of Syzygies
• - An Introduction to Markov Processes
• - Combinatorics of Coxeter Groups
• - An Introduction to Number Theory
• - Topics in Banach Space Theory
• - Analysis and Probability — Wavelets, Signals, Fractals
• - Compact Lie Groups
• - Bounded Analytic Functions
• - An Introduction to Operators on the Hardy-Hilbert Space
• - A Course in Enumeration
• - Number Theory — Volume I: Tools and Diophantine Equations
• - Number Theory — Volume II: Analytic and Modern Tools
• - The Arithmetic of Dynamical Systems
• - Abstract Algebra
• - Topological Methods in Group Theory
• - Graph Theory
• - Complex Analysis: Introduced in the Spirit of Lipman Bers
• - A Course in Commutative Banach Algebras
• - Braid Groups
• - Buildings Theory and Applications
• - Classical Fourier Analysis
• - Modern Fourier Analysis
• - The Finite Simple Groups
• - Distributions and Operators
• - Elementary Functional Analysis
• - Algebraic Function Fields and Codes
• - Symmetry, Representations, and Invariants
• - A Course in Commutative Algebra
• - Deformation Theory
• - Foundations of Optimization in Finite Dimensions
• - Ergodic Theory - with a view towards Number Theory
• - Monomial Ideals
• - Probability and Stochastics
• - Essentials of Integration Theory for Analysis
• - Analysis on Fock Spaces
• - Functional Analysis, Calculus of Variations and Optimal Control
• - Unbounded Self-adjoint Operators on Hilbert Space
• - Calculus Without Derivatives
• - Quantum Theory for Mathematicians
• - Geometric Analysis of the Bergman Kernel and Metric
• - Locally Convex Spaces
• - Fundamentals of Algebraic Topology
• - Integer Programming
• - Operator Theoretic Aspects of Ergodic Theory
• - Homotopical Topology
• - Brownian Motion, Martingales, and Stochastic Calculus
• - Differential Geometry - Connections, Curvature, and Characteristic Classes
• - Functional Analysis, Spectral Theory, and Applications
• - The Moment Problem
• - Modern Real Analysis
• - Binomial Ideals
• - Introduction to Real Analysis
• - Intersection Homology & Perverse Sheaves with Applications to Singularities
• - Measure, Integration & Real Analysis
• - Basic Representation Theory of Algebras
• - Spectral Theory - Basic Concepts and Applications
• - An Invitation to Unbounded Representations of Algebras on Hilbert Space
• - Lectures on Convex Geometry
• - Explorations in Complex Functions
• - Quaternion Algebras
• - Ergodic Dynamics - From Basic Theory to Applications
• - Lessons in Enumerative Combinatorics
• - Mathematical Logic
• - Random Walk, Brownian Motion and Martingales

これらの書籍は、GTMシリーズのほんの一部であり、数学の各分野における重要なトピックをカバーしています。GTMシリーズは、大学院レベルの数学教育において不可欠な存在であり、数学を深く理解するための基礎となる知識とスキルを提供します。

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